derivative of (2+x^{1/3}(1-x)^{2/3})

解

\frac{d}{d x} ((2 + x)^{\frac{1}{3}} (1 - x)^{\frac{2}{3}})

- \frac{x + 1}{( x + 2 ) ^{\frac{2}{3}} ( - x + 1 ) ^{\frac{1}{3}}}

解答ステップ

\frac{d}{d x} ((2 + x)^{\frac{1}{3}} (1 - x)^{\frac{2}{3}})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (2 + x)^{\frac{1}{3}}, g = (1 - x)^{\frac{2}{3}}

= \frac{d}{d x} ((2 + x)^{\frac{1}{3}}) (1 - x)^{\frac{2}{3}} + \frac{d}{d x} ((1 - x)^{\frac{2}{3}}) (2 + x)^{\frac{1}{3}}

\frac{d}{d x} ((2 + x)^{\frac{1}{3}}) = \frac{1}{3 ( 2 + x ) ^{\frac{2}{3}}}

\frac{d}{d x} ((1 - x)^{\frac{2}{3}}) = - \frac{2}{3 ( 1 - x ) ^{\frac{1}{3}}}

= \frac{1}{3 ( 2 + x ) ^{\frac{2}{3}}} (1 - x)^{\frac{2}{3}} + (- \frac{2}{3 ( 1 - x ) ^{\frac{1}{3}}}) (2 + x)^{\frac{1}{3}}

簡素化

\frac{1}{3 ( 2 + x ) ^{\frac{2}{3}}} (1 - x)^{\frac{2}{3}} + (- \frac{2}{3 ( 1 - x ) ^{\frac{1}{3}}}) (2 + x)^{\frac{1}{3}} : - \frac{x + 1}{( x + 2 ) ^{\frac{2}{3}} ( - x + 1 ) ^{\frac{1}{3}}}

= - \frac{x + 1}{( x + 2 ) ^{\frac{2}{3}} ( - x + 1 ) ^{\frac{1}{3}}}